https://file.aiquickdraw.com/mathbot/file/17262437832186mrdep01.jpg

The expression provided in the image is:

\[
2\sqrt{2a^3} - 16\sqrt{2b^3} + c^3 + 12abc
\]

To simplify this expression, let's break it down step by step.

### 1. Simplifying the Square Root Terms
- For \(2\sqrt{2a^3}\):
  - Recognize that \(a^3 = a^2 \cdot a\), so \( \sqrt{a^3} = \sqrt{a^2 \cdot a} = a\sqrt{a} \).
  - Thus, \( 2\sqrt{2a^3} = 2\sqrt{2} \cdot a\sqrt{a} \).

- For \(-16\sqrt{2b^3}\):
  - Similarly, \( \sqrt{b^3} = b\sqrt{b} \).
  - Thus, \( -16\sqrt{2b^3} = -16\sqrt{2} \cdot b\sqrt{b} \).

### 2. Combine All Terms
After simplifying, the expression can be rewritten as:

\[
2\sqrt{2} \cdot a\sqrt{a} - 16\sqrt{2} \cdot b\sqrt{b} + c^3 + 12abc
\]

### Final Expression
The expression in its simplified form is:

\[
2\sqrt{2} \cdot a\sqrt{a} - 16\sqrt{2} \cdot b\sqrt{b} + c^3 + 12abc
\]

Would you like to further explore any part of this expression or have any questions?

### Related Questions:
1. How would the expression change if \( a = b = c = 1 \)?
2. Can the expression be factored further?
3. What happens if we substitute \( a = 2 \), \( b = 3 \), and \( c = 4 \)?
4. How can we differentiate this expression with respect to \( a \)?
5. What is the significance of the square root in the context of this expression?

### Tip:
When simplifying expressions involving square roots, always look for opportunities to break down the terms inside the square root to see if any factors can be taken outside.

Simplify the expression: 2√2a³ - 16√2b³ + c³ + 12abc

Learn how to simplify the expression 2√2a³ - 16√2b³ + c³ + 12abc with a detailed step-by-step solution. This problem covers key algebraic concepts, including square roots and exponents. Suitable for students in Grades 9-12.

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